Find the volume of the region?

Question

Find the volume of the region enclosed by y=x+1, y=0, x=0, x=3 is rotated around the y-axis.

using calculus!!!!!

Answer 1

There is the hard way to do this, and there is the easy way. We're going to take the easy way. Notice that this is just a cylinder with a cone removed from it.

The cylinder is 4 high, with a radius of 3. Hence the volume is πr²h = 36π.

Now the volume of a cone/pyramid is Ah/3 where A is the area of the base and h is the height. In our case the cone has r=3 => A = 9π and h=3. Therefore, its volume is 9π.

Hence, the volume of the original solid is 36π - 9π = 27π

PS. Just to be clear, we are rotating a trapezoid around the y axis and the vertices of the trapezoid are (0,0), (0,1), (3,4) and (3,0).

Answer 2

This question is difficult without a diagram.
I think you will have to sketch a diagram as follows:-
E(2,0) , C(2,1) , B(2,3) , A(0,
The triangle ABC and square OECA are to be rotated about y axis.

Rectangle OECA in which the sides are OE = 2 and EC = 1:-
Rotating about y axis gives a cylinder of radius 2 and height 1
V1 = π x 2² x 1 = 4π units³

Rectangle ACBD in which sides AC= 2 and CB = 2
V2 = π x 2² x 2 = 8π units ³

Triangle ABD is rotated about y axis:-
V3 = ∫πx².dy between limits 1 to 3
V3 = π∫(y - 1)² dy
V3 = π∫(y² - 2y + 1) dy lims 1 to 3
V3 = π (y³/3 - y² + y) lims 1 to 3
V3 = π (3 - 1/3)
V3 = (2/3) π

Required Volume, V = V1 + V2 - V3
V = 4π + 8π - (2/3)π
V = 34π/3 units ³